Extra staff had been put in place to handle reports of abuse and from next month an in-tweet "report abuse" button, already available on some mobile devices, would be rolled out on the main Twitter website and apps.

"We want people to feel safe on Twitter, and we want the Twitter rules to send a clear message to anyone who thought that such behaviour was, or could ever be, acceptable," they wrote.

The announcement came after police revealed they were investigating allegations by eight people of abuse or threatening behaviour on the site. Scotland Yard said its e-crime unit was looking into the claims, three of which involved incidents outside London.

Twitter found itself in the spotlight after three female journalists, including the Guardian columnist Hadley Freeman, said they had been the subject of bomb threats on the site, while the Labour MP Stella Creasy and the campaigner Caroline Criado-Perez received threats of rape.

The threats provoked a backlash online, and a petition calling for Twitter to add a "report abuse" button to tweets attracted more than 120,000 signatures.

Twitter said it was working with the UK Safer Internet Centre, which promotes the safe and responsible use of technology.

In separate incidents, Creasy and Criado-Perez, who successfully fought for a woman's face to appear on £10 banknotes, were threatened on Twitter with rape. Two arrests have already been made in relation to those threats.

In a statement, the Metropolitan police said an investigation into eight allegations had been launched into all the claims.

"Detectives from the specialist organised and economic crime command have taken responsibility for the investigations into a number of allegations recently made to the MPS relating to allegations of malicious communication made on the social networking site Twitter.

"The police central e-crime unit (PCeU), who hold the police national cybercrime remit, is now investigating allegations made by eight people that they have been subject to harassment, malicious communication or bomb threats.

"Whilst outside PCeU's cyber operational remit, the MPS has taken the decision to centralise the individual investigations, including three that are outside London, to make the most effective use of resources avoid duplication."

The anonymous Twitter accounts from which the bomb threats originated were suspended.

Steve White, of the Police Federation, said on Saturday that the problem was "unpoliceable" and more needed to be done by social media organisations.

He told BBC Breakfast: "The organisations that run these social media platforms probably need to take a long hard look, they need to take some responsibility. It's much like when you go into a shop – there are prevention measures within shops, whether it be security guards or things locked away that you can't get to, which is going to prevent crime – and I think social media sites need to think long and hard about being able to prevent it from happening in the first place."